Question

If all the zero order correlation coefficients in a set of n-variates are equal to \(\rho\), then every third order partial correlation coefficient is equal to:

• A. \( \frac{2\rho}{1+\rho} \)
• B. \( \frac{\rho}{1+\rho} \)
• C. \( \frac{\rho}{1+3\rho} \)
• D. \( \rho \)

Hint:

For this specific problem where all zero-order correlations are equal to \(\rho\), you can memorize the general result: the \(k\)-th order partial correlation coefficient is given by \(r_k = \frac{\rho}{1+k\rho}\). This allows you to solve for any order instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves finding a general formula for higher-order partial correlation coefficients when all the initial (zero-order) correlations are identical. We can find the pattern by recursively applying the formula for partial correlation.

Step 2: Key Formula or Approach:
The recursive formula for a partial correlation coefficient is: \[ r_{12.3...k} = \frac{r_{12.3...(k-1)} - r_{1k.3...(k-1)} r_{2k.3...(k-1)}}{\sqrt{(1 - r^2_{1k.3...(k-1)})(1 - r^2_{2k.3...(k-1)})}} \] We will apply this starting from the first order and look for a pattern.

Step 3: Detailed Explanation:
Let all zero-order correlations be \(r_{ij} = \rho\). First-order partial correlation (e.g., \(r_{12.3}\)): \[ r_{12.3} = \frac{r_{12} - r_{13}r_{23}}{\sqrt{(1 - r^2_{13})(1 - r^2_{23})}} = \frac{\rho - \rho . \rho}{\sqrt{(1 - \rho^2)(1 - \rho^2)}} = \frac{\rho(1-\rho)}{1-\rho^2} = \frac{\rho(1-\rho)}{(1-\rho)(1+\rho)} = \frac{\rho}{1+\rho} \] So, all first-order partial correlations are equal to \(\rho_1 = \frac{\rho}{1+\rho}\). Second-order partial correlation (e.g., \(r_{12.34}\)): Using the recursive formula with the first-order partials: \[ r_{12.34} = \frac{r_{12.3} - r_{14.3}r_{24.3}}{\sqrt{(1 - r^2_{14.3})(1 - r^2_{24.3})}} = \frac{\rho_1 - \rho_1 . \rho_1}{1 - \rho_1^2} = \frac{\rho_1(1-\rho_1)}{(1-\rho_1)(1+\rho_1)} = \frac{\rho_1}{1+\rho_1} \] Substitute the value of \(\rho_1\): \[ r_{12.34} = \frac{\frac{\rho}{1+\rho}}{1 + \frac{\rho}{1+\rho}} = \frac{\frac{\rho}{1+\rho}}{\frac{1+\rho+\rho}{1+\rho}} = \frac{\rho}{1+2\rho} \] So, all second-order partial correlations are equal to \(\rho_2 = \frac{\rho}{1+2\rho}\). Third-order partial correlation (e.g., \(r_{12.345}\)): Using the same recursive logic, the third-order partial will be: \[ r_{12.345} = \frac{\rho_2}{1+\rho_2} \] Substitute the value of \(\rho_2\): \[ r_{12.345} = \frac{\frac{\rho}{1+2\rho}}{1 + \frac{\rho}{1+2\rho}} = \frac{\frac{\rho}{1+2\rho}}{\frac{1+2\rho+\rho}{1+2\rho}} = \frac{\rho}{1+3\rho} \] The general formula for the \(k\)-th order partial correlation coefficient is \( \frac{\rho}{1+k\rho} \).
Step 4: Final Answer:
Every third order partial correlation coefficient is equal to \( \frac{\rho}{1+3\rho} \).